Dissect & rearrange shapes
Theory
Cutting a shape into pieces and fitting them together a new way keeps the area the same, because nothing is added or removed. The perimeter can change, since different edges end up on the outside.
To dissect a shape is to cut it into pieces. To rearrange is to fit those pieces together a different way, with no gaps and no overlaps.
Because nothing is added and nothing is removed, the pieces still cover the same surface, so the area stays the same.
The perimeter can be different, because edges that were hidden inside may end up on the outside, and the other way round.
This idea is how the area formulas for triangles and parallelograms are built.
When pieces are rearranged, one measurement is fixed and one is not.
| Measurement | After rearranging |
|---|---|
| Area | always the same |
| Perimeter | may change |
| Number of pieces | does not affect the area |
How to reason about a rearranged shape
- Check the fit. The pieces must join with no gaps and no overlaps.
- Keep the area. The new shape covers exactly the same surface, so its area is unchanged.
- Re-measure the perimeter. Trace the new outline; it may be longer or shorter.
Nothing is added or removed, so it has the same area.
| \(3 \times 4\) | \(=\) | \(12\) |
The area stays the same. The perimeter may change.
| \(2 \times 6\) | \(=\) | \(12\) |
| \(3 \times 4\) | \(=\) | \(12\) |
The areas are equal.
Common pitfalls
Frequently asked questions
What does it mean to dissect a shape?
To dissect a shape is to cut it into pieces. Those pieces can then be rearranged into a new shape.
Does the area change when you rearrange the pieces?
No. As long as there are no gaps and no overlaps, the pieces cover the same surface, so the area stays exactly the same.
Does the perimeter stay the same too?
Not always. Rearranging can put different edges on the outside, so the perimeter of the new shape may be longer or shorter.
Why is this idea useful?
It is how area formulas are found. For example, a parallelogram can be cut and rearranged into a rectangle with the same area.
Does the number of pieces matter?
No. Whether a shape is cut into two pieces or ten, the total area of the rearranged shape is the same.